Most things that we observe in the world are of periodic nature, from the movements of the stars and the planets to our everyday life to the swaves and so on. On the other hand, starting with the work of Poincaré, we know that most dynamical systems are non-quasi-periodic (non-integrable), even chaotic . In this talk I want to explain this apparent paradox, and indicate some mathematical theories related to it.

Part 2. Torus actions in dynamical systems

In this talk, I want to show that any dynamical system admits intrinsic associated torus actions, and these actions play a very important role in many problems: local normalization, action-angle variables, perturbation theory, etc. There is a "new kind of conservation laws" for dynamical systems, which involves these torus actions.